I would like to establish some more examples of families of self-dual coherent algebras, in addition to the prototypical example of a family of self-dual coherent algebras arising from finite abelian groups that I established in my last post.

Suppose that \(\mathcal{W}\) and \(\mathcal{X}\) are commutative coherent algebras of order \(n\).

A duality mapping from \(\mathcal{W}\) to \(\mathcal{X}\) is a linear isomorphism \(T : \mathcal{W} \to \mathcal{X}\) such that:

  1. \(T(M N) = T(M) \circ T(N)\) for all \(M, N \in \mathcal{W}\).

  2. \(T(M \circ N) = \frac{1}{n} \ T(M) \ T(N)\) for all \(M, N \in \mathcal{W}\).

I would like to assume the fact that the tensor product of two commutative coherent algebras is self-dual if there is a duality mapping between them.

Then as a first additional example of a family of self-dual coherent algebras:

  • Suppose that \(G\) is a self-complementary strongly-regular graph of order \(n\).

    Assume that \(G\) is connected and \(k\)-regular.

    Then \(G\) has three distinct eigenvalues \(\{k, \lambda, -1 - \lambda \}\) for some \(\lambda \in \mathbb{R}\), since it is strongly-regular and self-complementary.

    Moreover:

    \[\mathbb{C}[A] := \operatorname{span} \{ I, A, A^{2}, \cdots \} = \operatorname{span} \left \{ I, A, P A P^{T} \right \} = \operatorname{span} \left \{ \frac{1}{n} J, E, P E P^{T} \right \}\]

    for some permutation matrix \(P \in \operatorname{Mat}_{n \times n}(\mathbb{C})\) and orthogonal projection \(E \in \operatorname{Mat}_{n \times n}(\mathbb{C})\) such that \(A = k \left (\frac{1}{n} J \right ) + \lambda E + (-1 - \lambda) P E P^{T}\), letting \(A \in \operatorname{Mat}_{n \times n}(\mathbb{C})\) denote the adjacency matrix of \(G\) and \(J\) denote the all-ones matrix in \(\operatorname{Mat}_{n \times n}(\mathbb{C})\).

    Now, let \(T : \mathbb{C}[A] \to \mathbb{C}[A]\) denote the linear automorphism on \(\mathbb{C}[A]\) such that \(T(E) := A\), \(T \left ( P E P^{T} \right ) := P A P^{T}\), and \(T \left ( \frac{1}{n} J \right ) := I\).

    \[\implies T \left ( A + P A P^{T} \right )\] \[= \left ( 2k \right ) \ T \left ( \frac{1}{n} J \right ) - T(E) - T \left (P E P^{T} \right )\] \[= (2k)I - A - P A P^{T}\] \[= (n-1)I - A - P A P^{T}\]

    [ again, since \(G\) is self-complementary ]

    \[= nI - \left ( I + A + P A P^{T} \right )\] \[= nI - J\] \[\implies T \left ( A + P A P^{T} \right ) = \frac{1}{n} \left ( T \left ( A + P A P^{T} \right ) \right )^{2}\]

    Observe now that \(\mathbb{C}[A]\) is a commutative coherent algebra and that \(T \left ( P M P^{T} \right ) = P \ T(M) \ P^{T}\) for all \(M \in \mathbb{C}[A]\).

    \[\implies \frac{1}{n} T(A) \in \left \{ E, P E P^{T} \right \}\] \[\implies T \left ( M \circ N \right ) = \frac{1}{n} T(M) \ T(N) \text { for all } M, N \in \mathbb{C}[A]\]

    It can then be concluded that \(\mathcal{C}[A] \otimes \mathcal{C}[A]\) is a self-dual coherent algebra, since \(T(M N) = T(M) \circ T(N)\) for all \(M,N \in \mathcal{C}[A]\) by construction as well.

So, self-dual coherent algebras can be constructed from e.g. self-complementary arc-transitive graphs in this way, for instance.

Now, a complex square matrix \(M\) (of order \(n\)) is said to be Schur-invertible with Schur-inverse \(M^{(-)}\) if \(M \circ M^{(-)} = J\) for some (unique) matrix \(M^{(-)} \in \operatorname{Mat}_{n \times n}(\mathbb{C})\) and all-ones matrix \(J \in \operatorname{Mat}_{n \times n}(\mathbb{C})\).

The Nomura algebra \(\mathcal{N}_{M}\) (of order \(n\)) of a Schur-invertible matrix \(M \in \operatorname{Mat}_{n \times n}(\mathbb{C})\) is defined as \(\mathcal{N}_{M} := \{ N \in \operatorname{Mat}_{n \times n}(\mathbb{C}) : N \left ( M e_{i} \circ M^{(-)} e_{j} \right ) \in \operatorname{span} \left \{ M e_{i} \circ M^{(-)} e_{j} \right \} \text{ for all } 1 \leq i,j \leq n \}\); the Nomura algebra of Schur-invertible matrix is a subspace of complex square matrices that is closed under ordinary matrix multiplication with identity.

A Schur-invertible matrix \(M\) of order \(n\) is said to be Type-II if \(M M^{(-)} = n I\).

I would like to construct a second example of a family of self-dual coherent algebras from Nomura algebras of Type-II matrices:

  1. Let \(\Lambda_{M} : \mathcal{N}_{M} \to \operatorname{Mat}_{n \times n}(\mathbb{C})\) denote the linear mapping such that \(N \left ( M e_{i} \ \circ \ M e_{j} \right ) = \left ( \Lambda_{M} (N) \right )_{i,j} \ \left ( M e_{i} \ \circ \ M e_{j} \right )\) for all \(1 \leq i, j \leq n\) and \(N \in \mathcal{N}_{M}\), for any Schur-invertible matrix \(M\) (of order \(n\)).

  2. Suppose that \(M\) is a Type-II matrix of order \(n\).

    Select any \(N \in \mathcal{N}_{M}\).

    Then for all \(1 \leq i, j, k \leq n\):

    \[\implies e_{k}^{T} \left ( \Lambda_{M}(N) \left ( M^{T} e_{i} \circ M^{(-)T} e_{j} \right ) \right )\] \[= \sum\limits_{l = 1}^{n} \left ( \Lambda_{M}(N) \right )_{k, l} \ M_{i, l} \ M_{j,l}^{(-)}\] \[= \sum\limits_{l = 1}^{n} \left ( \sum\limits_{m = 1}^{m} \left ( N_{j,m} M_{m,k} M_{m,l}^{(-)} \right ) \left ( M_{j,k}^{(-)} M_{j, l} \right ) \right ) M_{i,l} M_{j,l}^{(-)}\] \[= M_{j,k}^{(-)} \left ( \sum\limits_{m = 1}^{n} \left ( N_{j,m} M_{m,k} \right ) \left ( \sum\limits_{l = 1}^{m} M_{i, l} M_{m, l}^{(-)} \right ) \right )\] \[= M_{j,k}^{(-)} \left ( N_{j, i} M_{i,k} \right ) \left ( n \right )\]

    [ since \(M\) is Type-II ]

    \[= \left ( n N_{j, i} \right ) \left ( e_{k}^{T} \left ( M^{T} e_{i} \circ M^{(-){T}} e_{j} \right) \right )\]

    .

    \[\implies \Lambda_{M}(N) \in \mathcal{N}_{M^{T}}\]

    [ noting that \(M^{T}\) is also Type-II ].

    By similar reasoning, it follows that \(\Lambda_{M^{T}} \left ( \frac{1}{n} \ \Lambda_{M}(N) \right ) = N^{T} \in \mathcal{N}_{M}\).

    It is then ultimately apparent that \(\mathcal{N}_{M}\) is closed under transposition in general.

    Observe moreover that \(\Lambda_{M} : \mathcal{N}_{M} \to \mathcal{N}_{M_{T}}\) is in fact a linear isomorphism (with inverse \(\left ( \Lambda_{M^{T}} \bullet \Lambda_{M} \right )^{2}\)): hence it is ultimately apparent too that \(\mathcal{N}_{M}\) is closed under Schur-product (since \(\mathcal{N}_{M^{T}}\) is closed under ordinary matrix multiplication).

    Since \(M\) is Type-II, it immediately follows that the all-ones matrix \(J \in \operatorname{Mat}_{n \times n}(\mathbb{C})\) is in \(\mathcal{N}_{M}\) as well.

    Hence it can be established here that \(\mathcal{N}_{M}\) is a coherent algebra; indeed, \(\mathcal{N}_{M}\) is a commutative coherent algebra since \(X Y = \Lambda_{M}^{-1} \left ( \Lambda_{M}(X) \circ \Lambda_{M}(Y) \right ) = \Lambda_{M}^{-1} \left ( \Lambda_{M}(Y) \circ \Lambda_{M}(X) \right ) = Y X\) for all \(X, Y \in \mathcal{N}_{M}\).

    It can then be similarly seen that \(\mathcal{N}_{M^{T}}\) is a commutative coherent algebra as well.

    It is then apparent that \(\Lambda_{M} : \mathcal{N}_{M} \to \mathcal{N}_{M^{T}}\) is a duality mapping between commutative coherent algebras \(\mathcal{N}_{M}\) and \(\mathcal{N}_{M^T}\): since \(\frac{1}{n} \left ( \Lambda_{M^{T}} \bullet \Lambda_{M} \right )\) maps Schur-primitive matrices in \(\mathcal{N}_{M}\) to their Schur-primitive transpose in \(\mathcal{N}_{M}\), it is necessary that \(\frac{1}{n} \Lambda_{M}\) maps Schur-primitive matrices in \(\mathcal{N}_{M}\) to primitive matrices in \(\mathcal{N}_{M^{T}}\) and moreover that \(\Lambda_{M}(X \circ Y) = \frac{1}{n} \Lambda_{M}(X) \ \Lambda_{M}(Y)\) for all \(X, Y \in \mathcal{N}_{M}\).

    Finally, it can then be concluded here that the tensor product \(\mathcal{N}_{M} \otimes \mathcal{N}_{M^{T}}\) of Nomura algebras \(\mathcal{N}_{M}\) and \(\mathcal{N}_{M^{T}}\) for the Type-II matrix \(M\) is a self-dual coherent algebra.

So, self-dual coherent algebras can indeed be constructed from Type-II matrices; complex Hadamard matrices are a first example of a family of Type-II matrices that comes to mind, which moreover can e.g. be constructed from character tables of finite abelian groups and graphs that admit uniform mixing (with respect to their continuous-time quantum walks relative to their adjacency matrices).

I would now lastly like to establish that the adjacency algebras of Hamming graphs are self-dual coherent algebras:

  • Suppose that \(G := H^{n}\) is the \(n\)-th direct power of some finite cyclic group \(H\) and \(n \in \mathbb{N}\).

    Let \(\mathcal{W}_{G}\) denote the coherent algebra arising from \(G\) as in my previous posts: i.e. \(\mathcal{W}_{G} := \operatorname{span} \{ A_{g} : g \in G \}\) where \(A_{g} \in \operatorname{Mat}_{G \times G}(\mathbb{C})\) is a \(01\)-matrix relating pairs of elements of \(G\) whose difference is \(g\) (and hence also a permutation matrix representation the left-translation of the elements of \(G\) by \(g\)) for all \(g \in G\).

    Additionally, let \(f_{G} : G \to \{ 0, \cdots, n \}\) denote a Hamming weight function on \(G\): i.e. a function mapping elements of \(G\) to their number of non-zero coordinates.

    Partition the elements of \(G\) according to their Hamming weight: define \(G_{j} := \{ g \in G : f_{G}(g) = j \}\) for all \(0 \leq j \leq n\).

    For all \(0 \leq j \leq n\), let \(A_{j} \in \operatorname{Mat}_{G \times G}(\mathbb{C})\) denote the \(01\)-matrix in \(\in \operatorname{Mat}_{G \times G}(\mathbb{C})\) relating pairs of elements of \(G\) whose difference has Hamming weight \(j\):

    i.e. \(\left ( A_{j} \right )_{x,y} := \begin{cases} 1 & \text{ if } f_{G}(xy^{-1}) = j \\ 0 & \text{ otherwise} \end{cases}\) for all \(x, y \in G\).

    Define \(\mathcal{X}_{G} := \operatorname{span} \{ A_{j} : 0 \leq j \leq n \} \subseteq \operatorname{Mat}_{G \times G}(\mathbb{C})\).

    Note that \(\mathcal{X}_{G} \subseteq \mathcal{W}_{G}\), since \(G\) is abelian and hence \(f_{G}\left ( \left ( z x \right ) \left ( z y \right )^{-1} \right ) = f_{G} \left ( x y^{-1} \right )\) for all \(x,y,z \in G\) since \(G\) is abelian.

    It is then apparent that \(\mathcal{X}_{G}\) is a commutative coherent subalgebra of \(\mathcal{W}_{G}\), observing that \(\mathcal{X}_{G}\) can be identified with the adjacency algebra of a Hamming graph (which is a commutative coherent algebra, assuming the fact that Hamming graphs are distance-regular).

    Now, recall too from a previous post that the set \(\{ \chi_{g} : g \in G \}\) of irreducible characters of \(G\) can be indexed in such a way that \(\chi_{x}(y) = \chi_{y}(x)\) for all \(x, y \in G\) (and that \(\mathcal{W}_{G}\) is self-dual under the character table of \(\mathcal{W}_{G}\) relative to this way of indexing the irreducible characters of \(G\)).

    Observe too then that \(\sum\limits_{g \in G_{j}} \chi_{g}(h)\) depends only on the Hamming weight of \(h\) for all \(h \in G\) and all \(0 \leq j \leq n\), which essentially follows from the the fact that \(G\) is a direct power of a finite cyclic group.

    Define the matrix \(P \in \operatorname{Mat}_{n \times n}(\mathbb{C})\) as \(P_{j,k} := \frac{1}{\left\vert G_{k} \right\vert} \left ( \sum\limits_{(g, h) \in G_{j} \times G_{k}} \overline{\chi_{g}(h)} \right )\) for all \(0 \leq j, k \leq n\).

    From previous posts, it then follows \(P\) is character table for \(\mathcal{X}_{G}\).

    Note too that \(P = \overline{P}\), since \(\mathcal{X}_{G}\) is the adjacency algebra of distance-regular graph and hence every Schur-primitive matrix in \(\mathcal{X}_{G}\) is a Hermitian matrix (whose distinct eigenvalues are all real).

    Lastly, for all \(0 \leq j \leq n\), define \(E_{j} \in \operatorname{Mat}_{G \times G}(\mathbb{C})\) as \(E_{j} := \frac{1}{\left\vert G \right\vert} \ F_{j}^{*} F_{j}\) where \(F_{j} \in \operatorname{Mat}_{G_{j} \times G}(\mathbb{C})\) is defined as \(\left ( F_{j} \right )_{x,y} := \overline{\chi_{x}(y)}\) for all \(x \in G_{j}\) and \(y \in G\).

    It then be observed that \(A_{j} = \sum\limits_{k = 0}^{n} P_{j,k} E_{k}\) and that \(E_{j} = \sum\limits_{k = n} P_{j,k} A_{k}\) for all \(0 \leq j \leq n\).

    Moreover, it follows from the fact the irreducible characters of \(G\) form an orthonormal basis for the vector space \(L^{2}(G)\) of complex-valued functions on \(G\) that each \(E_{j}\) is idempotent for all \(0 \leq j \leq n\).

    Hence it can be concluded that \(\{ E_{j} : 0 \leq j \leq n \} \subseteq \mathcal{X}_{G}\) is the set of primitive matrices in \(\mathcal{X}_{G}\) and that \(\mathcal{X}_{G}\) is a self-dual coherent algebra.

It turns out that, conversely, the adjacency algebra of every Hamming graph can be identified with an instance of this kind of coherent algebra arising from a finite cyclic group as well; hence the adjacency algebras of Hamming graphs are indeed self-dual.

So, indeed there are some arguably interesting families of self-dual coherent algebras that can be constructed aside from the prototypical example of a family of self-dual coherent algebras arising from finite abelian groups that I established in my previous post. I hope to continue to e.g. construct even more examples of self-dual coherent algebras in future posts towards e.g. completely classifying certain families of self-dual coherent algebras, such as certain families of distance-regular graphs whose adjacency algebras are self-dual.